Abstract
A particular weighted generalization of some classical zero-sum constants was first considered about fifteen years back. Since then, many people got interested in this generalization. Finding exact values of these constants for the finite cyclic group \({{\mathbb {Z}}}_n\) has been one of the interesting problems and people have worked on them for various weights. After summarizing the important known results on the weighted Davenport constant of a finite cyclic group for various weight sets, here we take up the problem of determining the exact values and providing bounds of it corresponding to some new weight sets.
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References
Adhikari S D, Balasubramanian R, Pappalardi F and Rath P, Some zero-sum constants with weights, Proc. Indian Acad. Sci. (Math. Sci.) 118 (2008) 183–188
Adhikari S D and Chen Y G, Davenport constant with weights and some related questions—II, J. Combin. Theory Ser. A 115 (2008) 178–184
Adhikari S D, Chen Y G, Friedlander J B, Konyagin S V and Pappalardi F, Contributions to zero-sum problems, Discrete Math. 306 (2008) 1–10
Adhikari S D, David C and Urroz J J, Generalizations of some zero-sum theorems, Integers 8 (2008) paper A52
Adhikari S D and Mazumdar E, Modifications of some methods in the study of zero-sum constants, Integers 14 (2014) paper A25
Adhikari S D and Rath P, Davenport constant with weights and some related questions, Integers 6 (2006) A30
Chintamani M N and Moriya B K, Generalizations of some zero sum theorems, Proc. Indian Acad. Sci. (Math. Sci.) 122(1) (2012) 15–21
Gao W D, A combinatorial problem on finite abelian groups, J. Number Theory 58 (1996) 100–103
Geroldinger A and Halter-Koch F, Non-unique factorizations: algebraic, combinatorial and analytic theory, pure and applied mathematics (Boca Raton), vol. 278 (2006) (Boca Raton, FL: Chapman & Hall/CRC)
Griffiths S, The Erdős–Ginzburg–Ziv theorem with units, Discrete Math. 308(23) (2008) 5473–5484
Grynkiewicz D J and Hennecart F, A weighted zero-sum problem with quadratic residues, Unif. Distrib. Theory 10(1) (2015) 69–105
Grynkiewicz D J, Marchan L E and Ordaz O, A weighted generalization of two theorems of Gao, Ramanujan J. 28(3) (2012) 323–340
Kemperman J H B and Scherk P, Complexes in abelian groups, Can. J. Math. 6 (1954) 230–237
Luca F, A generalization of a classical zero-sum problem, Discrete Math. 307(13) (2007) 1672–1678
Scherk P, Solution to Problem 4466, Amer. Math. Monthly 62 (1955) 46–47
Thangadurai R, A variant of Davenport’s constant, Proc. Indian Acad. Sci. (Math. Sci.) 117(2) (2007) 147–158
Xia X, Two generalized constants related to zero-sum problems for two special sets, Integers 7 (2007) paper A52
Xia X and Li Z, Some Davenport constants with weights and Adhikari and Rath’s conjecture, Ars Combin. 88 (2008) 83–95
Yuan P and Zeng X, Davenport constant with weights, Eur. J. Combin. 31 (2010) 677–680
Acknowledgements
The first author would like to acknowledge a fellowship (under MATRICS) from SERB, Department of Science & Technology, Government of India, and the second author would like to acknowledge CSIR, Government of India, for a research fellowship.
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Communicated by Sanoli Gun.
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Adhikari, S.D., Hegde, S. Zero-sum constants involving weights. Proc Math Sci 131, 37 (2021). https://doi.org/10.1007/s12044-021-00634-7
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DOI: https://doi.org/10.1007/s12044-021-00634-7